Question

arcsin(x-1)^2积分

f(x)的导数为arcsin(x-1)^2,f(0)=0,求函数f(x)在区间（0,1）上的几分
只要给出具体思路即可，

Answer 1

简单计算一下即可，答案如图所示

Answer 2

f’(x)=arcsin(x-1)^2
先求出f(x)=∫arcsin(x-1)^2dx,由于f(0)=0,f(x)=∫(0,x)arcsin(y-1)^2dy
然后对f(x)进行积分：
∫(0,1)f(x)dx
=∫(0,1)dx∫(0,x)arcsin(y-1)^2dy (交换积分顺序）
=∫(0,1)dy∫(y,1)arcsin(y-1)^2dx
=∫(0,1)(1-y)arcsin(y-1)^2dy
=(-1/2)∫(0,1)arcsin(y-1)^2d(y-1)^2 (用公式∫arcsinxdx)
=(-1/2)[(y-1)^2arcsin(y-1)^2+√(1-(y-1)^4)] | (0,1)
=(-1/2)(1-π/2)
=(π-2)/4